reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;
reserve P,Q,R for POINT of IncProjSp_of real_projective_plane,
            L for LINE of IncProjSp_of real_projective_plane,
        p,q,r for Point of real_projective_plane;
reserve u,v,w for non zero Element of TOP-REAL 3;

theorem
  for a,b,c,d,e,f being Real for u1,u2 being non zero Element of TOP-REAL 3
  st Dir u1 = Dir u2 & qfconic(a,b,c,d,e,f,u1) is negative holds
  qfconic(a,b,c,d,e,f,u2) is negative
  proof
    let a,b,c,d,e,f be Real;
    let u1,u2 be non zero Element of TOP-REAL 3;
    assume that
A1: Dir u1 = Dir u2 and
A2: qfconic(a,b,c,d,e,f,u1) is negative;
    are_Prop u2,u1 by A1,ANPROJ_1:22;
    then consider g be Real such that
A3: g <> 0 and
A4: u2 = g * u1 by ANPROJ_1:1;
A5: u2.1 = g * u1.1 & u2.2 = g * u1.2 & u2.3 = g * u1.3 by A4,RVSUM_1:44;
    0 < g^2 by A3,SQUARE_1:12;
    then reconsider g2 = g * g as positive Real;
    qfconic(a,b,c,d,e,f,u2) = a * u2.1 * u2.1 + b * u2.2 * u2.2
                               + c * u2.3 * u2.3 + d * u2.1 * u2.2
                               + e * u2.1 * u2.3 + f * u2.2 * u2.3
                                 by PASCAL:def 1
                           .= g2 * (a * u1.1 * u1.1 + b * u1.2 * u1.2
                             + c * u1.3 * u1.3 + d * u1.1 * u1.2
                             + e * u1.1 * u1.3 + f * u1.2 * u1.3) by A5
                           .= g2 * qfconic(a,b,c,d,e,f,u1) by PASCAL:def 1;
    hence thesis by A2;
  end;
